Question: The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T).  Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible?
Solution: We consider the opposite; we try to find the number of words that do not contain A, and then subtract it from the total possible number of words. So we have a few cases to consider:

$\bullet$  One letter words: There is only $1$ one-letter word that contains A, that's A.

$\bullet$  Two letter words: There are $19\times19=361$ words that do not contain A. There is a total of $20\times20=400$ words, so we have $400-361=39$ words that satisfy the condition.

$\bullet$  Three letter words: There are $19\times19\times19=6859$ words without A, and there are $20^{3}=8000$ words available. So there are $8000-6859=1141$ words that satisfy the condition.

$\bullet$  Four letter words: Using the same idea as above, we have $20^{4}-19^{4}=29679$ words satisfying the requirement.

So this gives a total of $1+39+1141+29679=\boxed{30860}$ words.